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We consider the approximate recovery of multivariate periodic functions from a discrete set of function values taken on a rank-1 lattice. Moreover, the main result is the fact that any (non-)linear reconstruction algorithm taking function values on any integration lattice of size M has a dimension-independent lower bound of $$2^{-(\\alpha +1)/2} M^{-\\alpha /2}$$2-(¿+1)/2M-¿/2 when considering the optimal worst-case error with respect to function spaces of (hybrid) mixed smoothness $$\\alpha 0$$¿0 on the d-torus. We complement this lower bound with upper bounds that coincide up to logarithmic terms. These upper bounds are obtained by a detailed analysis of a rank-1 lattice sampling strategy, where the rank-1 lattices are constructed by a component---by---component method. The lattice (group) structure allows for an efficient approximation of the underlying function from its sampled values using a single one-dimensional fast Fourier transform. This is one reason why these algorithms keep attracting significant interest. We compare our results to recent (almost) optimal methods based upon samples on sparse grids.

Tight error bounds for rank-1 lattice sampling in spaces of hybrid mixed smoothness